Question

Teams in the National Football League (NFL) in the US play four pre-season games each year before the regular season starts. Do teams that do well in the pre-season tend to also do well in the regular season? We are interested in whether there is a positive linear association between the number of wins in the pre-season and the number of wins in the regular season for teams in the NFL. (a) What are the null and alternative hypotheses for this test? (b) The correlation between these two variables for the 32 NFL teams over the 10 year period from 2005 to 2014 is 0.067 . Use this sample (with \(n=320\) ) to calculate the appropriate test statistic and determine the p-value for the test. (c) State the conclusion in context, using a \(5 \%\) significance level. (d) When an NFL team goes undefeated in the pre-season, should the fans expect lots of wins in the regular season?

Short answer

(a) H0: There is no linear correlation between pre-season wins and regular season wins. Ha: There is a positive linear correlation between pre-season wins and regular season wins. (b) A calculation of the test statistic and the p-value is required here; which usually involves a statistical software/tool. (c) The conclusion is drawn based on whether the p-value is less than the significance level, interpreted within the context of NFL games. (d) The final interpretation on expectations from an undefeated pre-season is drawn in consideration with the results of the hypothesis test.

Step-by-step solution

Formulate the Null and Alternative hypothesis

For part (a), the null hypothesis (H0) and the alternative hypothesis (Ha) would be formulated based on the assumption of no relationship and the possibility of a relationship respectively. H0: There is no linear correlation between pre-season wins and regular season wins. Ha: There is a positive linear correlation between pre-season wins and regular season wins.

Calculating the Test Statistic

In part (b), knowing the correlation (r=0.067) and the sample size (n=320), we calculate the test statistic (t) as follows: The formula is \(t = r \sqrt{(n-2)/(1-r^2)}\). Plugging the values of \(r\) and \(n\) into this equation, we calculate \(t\).

Determine the p-value

The p-value would be calculated using the generated test statistic with a degrees of freedom of n-2. Typically, this would require statistical software or lookup tables.

Conclusion in context

In step (c), the p-value is weighed against the significance level (in this case, 5%). If the p-value is lower than the significance level, we reject the null hypothesis, suggesting a significant correlation. The conclusion would then be contextualized to the NFL setting.

Interpretation

In step (d), based on the results, an interpretation of the effect of an undefeated pre-season on fans' expectations would be made. The conclusion would be drawn based on the p-value indicating a significant or insignificant relationship.